3.437 \(\int e^x \sin ^{-1}(e^x) \, dx\)

Optimal. Leaf size=22 \[ \sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]

[Out]

Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]

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Rubi [A]  time = 0.038355, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2194, 4844, 2246, 32} \[ \sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x*ArcSin[E^x],x]

[Out]

Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 4844

Int[((a_.) + ArcSin[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcSin[u], w, x] - Dist
[b, Int[SimplifyIntegrand[(w*D[u, x])/Sqrt[1 - u^2], x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}
, x] && InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int e^x \sin ^{-1}\left (e^x\right ) \, dx &=e^x \sin ^{-1}\left (e^x\right )-\int \frac{e^{2 x}}{\sqrt{1-e^{2 x}}} \, dx\\ &=e^x \sin ^{-1}\left (e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x}} \, dx,x,e^{2 x}\right )\\ &=\sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0082238, size = 22, normalized size = 1. \[ \sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*ArcSin[E^x],x]

[Out]

Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]

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Maple [A]  time = 0.009, size = 18, normalized size = 0.8 \begin{align*}{{\rm e}^{x}}\arcsin \left ({{\rm e}^{x}} \right ) +\sqrt{- \left ({{\rm e}^{x}} \right ) ^{2}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*arcsin(exp(x)),x)

[Out]

exp(x)*arcsin(exp(x))+(-exp(x)^2+1)^(1/2)

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Maxima [A]  time = 1.4467, size = 23, normalized size = 1.05 \begin{align*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt{-e^{\left (2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*arcsin(exp(x)),x, algorithm="maxima")

[Out]

arcsin(e^x)*e^x + sqrt(-e^(2*x) + 1)

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Fricas [A]  time = 2.00802, size = 51, normalized size = 2.32 \begin{align*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt{-e^{\left (2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*arcsin(exp(x)),x, algorithm="fricas")

[Out]

arcsin(e^x)*e^x + sqrt(-e^(2*x) + 1)

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Sympy [A]  time = 0.829462, size = 17, normalized size = 0.77 \begin{align*} \sqrt{1 - e^{2 x}} + e^{x} \operatorname{asin}{\left (e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*asin(exp(x)),x)

[Out]

sqrt(1 - exp(2*x)) + exp(x)*asin(exp(x))

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Giac [A]  time = 1.15127, size = 23, normalized size = 1.05 \begin{align*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt{-e^{\left (2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*arcsin(exp(x)),x, algorithm="giac")

[Out]

arcsin(e^x)*e^x + sqrt(-e^(2*x) + 1)